Definition: In this task, you are given a second sentence. Your task is to generate the first sentence on the same topic but incoherent and inconsistent with the second sentence.
Input: Therefore , the remainder of this article will be restricted to the quotients of polynomial rings by ideals .
Output:
Thus , the graded algebras generated by elements of degree 1 are exactly , up to an isomorphism , the quotients of polynomial rings by homogeneous ideals .